Proyecciones
Vol. 22, N o 1, pp. 1-13, May 2003.
Universidad Católica del Norte
Antofagasta - Chile

NONRESONANCE BELOW THE SECOND
EIGENVALUE FOR A NONLINEAR
ELLIPTIC PROBLEM

M. MOUSSAOUI and M. MOUSSAOUI
University Mohamed I, Oujda, Morocco

Abstract

We study the solvability of the problem

when the nonlinearity g is assumed to lie asymptotically between 0 and the second eigenvalue ¸ We show that this problem is nonresonant.

References

[1] R. A. ADAMS Sobolev spaces, Academic Press, New York, (1975)

[2] A. ANANE Simplicité et isolation de la première valeur propre du p-laplacien avec poids C. R. Ac. Sc. Paris, 305 (1987), 725 ¡ 728.

[3] A. Anane, O. Chakrone Sur un théorème de point critique et application à un problème de non résonance entre deux valeurs propres du p-laplacien. Annales de la Faculté des Sciences de Toulouse, Vol IX,N^o 1, 2000, p.5-30.

[4] A. ANANE, N. TSOULI On the second eigenvalue of the p-laplacian, Nonlinear Partial Differential Equations, Pitman Research Notes 343,1-9 (1996)

[5] BARTOLO. P, BENCI. D. FORTUNATO Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Analysis 7, 981-1012 (1983).

[6] D. G. COSTA, A. S. OLIVEIRA Existence of solution for a class of semilinear elliptic problems at double resonance Bol. Soc. BRAS. Mat., vol 19, (1988) 21-37.

[7] D. G. COSTA, C. A. MAGALHAES Variational elliptic problems which are nonquadratic at infinity. Nonlinear Analysis, vol 23. No 11. 1401-1412 (1994).

[8] D.G. DE FIGUEIREDO, J.P. GOSSEZ Strict monotonicity of eigenvalues and unique continuation, Comm. Part. Diff. Eq., 17, 339-346 (1992)

[9] M. MOUSSAOUI, A.R. EL AMROUSS Minimax principles for critical- point theory in application to quasilinear boundary-value problems E.J.D.E vol (2000) N 18 p 1-9.

[10] M. MOUSSAOUI, M. MOUSSAOUI Nonlinear elliptic problems with resonance at the two first eigenvalue: A variational approach , preprint (2000).

[11] M. MOUSSAOUI, M. MOUSSAOUI Nonresonance between non-consecutive eigenvalues of semilinear elliptic equations: Variational methods, preprint (2000).

[12] JOAO. MARCOS. B. do Ó Solution to perturbed eigenvalue problems of the p-laplacian in R^N *J. D. E. N 11, 1-15 (1997).

[13] P. H. RABINOWITZ minimax methods in critical point theory with application to differential equations, CBMS, Regional conf. Ser. Math., vol 65 AMS, Providence Ri. (1986).

[14] E. A. B. SILVA Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Analysis TMA, 16, 455-477 (1991).

Received : May 2002.

M. MOUSSAOUI
Department of Mathematics
Faculty of Sciences
University Mohamed I
P. O. BOX 524
60000 Oujda
Morocco
e-mail : moussaoui@sciences.univ-oujda.ac.ma

and

M. MOUSSAOUI
Department of Mathematics
Faculty of Sciences
University Mohamed I
P. O. BOX 524
60000 Oujda
Morocco
e-mail : mimoun moussaoui@hotmail.com